src/Provers/Arith/combine_numerals.ML
 author paulson Fri Jun 16 13:13:55 2000 +0200 (2000-06-16) changeset 9073 40d8dfac96b8 parent 8816 31b4fb3d8d60 child 9191 300bd596d696 permissions -rw-r--r--
tracing flag for arith_tac
1 (*  Title:      Provers/Arith/combine_numerals.ML
2     ID:         \$Id\$
3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
4     Copyright   2000  University of Cambridge
6 Combine coefficients in expressions:
8      i + #m*u + j ... + #n*u + k  ==  #(m+n)*u + (i + (j + k))
10 It works by (a) massaging the sum to bring the selected terms to the front:
12      #m*u + (#n*u + (i + (j + k)))
14 (b) then using left_distrib to reach
16      #(m+n)*u + (i + (j + k))
17 *)
19 signature COMBINE_NUMERALS_DATA =
20 sig
21   (*abstract syntax*)
22   val mk_sum: term list -> term
23   val dest_sum: term -> term list
24   val mk_coeff: int * term -> term
25   val dest_coeff: term -> int * term
26   (*rules*)
27   val left_distrib: thm
28   (*proof tools*)
29   val prove_conv: tactic list -> Sign.sg -> term * term -> thm option
30   val trans_tac: thm option -> tactic (*applies the initial lemma*)
31   val norm_tac: tactic                (*proves the initial lemma*)
32   val numeral_simp_tac: tactic        (*proves the final theorem*)
33   val simplify_meta_eq: thm -> thm    (*simplifies the final theorem*)
34 end;
37 functor CombineNumeralsFun(Data: COMBINE_NUMERALS_DATA):
38   sig
39   val proc: Sign.sg -> thm list -> term -> thm option
40   end
41 =
42 struct
44 (*Remove the first occurrence of #m*u from the term list*)
45 fun remove (_, _, []) = (*impossible, since #m*u was found by find_repeated*)
46       raise TERM("combine_numerals: remove", [])
47   | remove (m, u, t::terms) =
48       let val (n,v) = Data.dest_coeff t
49       in  if m=n andalso u aconv v then terms
50           else t :: remove (m, u, terms)
51       end;
53 (*a left-to-right scan of terms, seeking another term of the form #n*u, where
54   #m*u is already in terms for some m*)
55 fun find_repeated (tab, _, []) = raise TERM("find_repeated", [])
56   | find_repeated (tab, past, t::terms) =
57       let val (n,u) = Data.dest_coeff t
58       in
59 	  case Termtab.lookup (tab, u) of
60 	      Some m => (u, m, n, rev (remove (m,u,past)) @ terms)
61 	    | None => find_repeated (Termtab.update ((u,n), tab),
62 				     t::past,  terms)
63       end;
65 (*the simplification procedure*)
66 fun proc sg _ t =
67   let val (u,m,n,terms) = find_repeated (Termtab.empty, [], Data.dest_sum t)
68       val reshape =  (*Move i*u to the front and put j*u into standard form
69 		       i + #m + j + k == #m + i + (j + k) *)
70 	    if m=0 orelse n=0 then   (*trivial, so do nothing*)
71 		raise TERM("combine_numerals", [])
72 	    else Data.prove_conv [Data.norm_tac] sg
73 			(t,
74 			 Data.mk_sum ([Data.mk_coeff(m,u),
75 				       Data.mk_coeff(n,u)] @ terms))
76   in
77       apsome Data.simplify_meta_eq
78 	 (Data.prove_conv
79 	    [Data.trans_tac reshape, rtac Data.left_distrib 1,
80 	     Data.numeral_simp_tac] sg
81 	    (t, Data.mk_sum (Data.mk_coeff(m+n,u) :: terms)))
82   end
83   handle TERM _ => None
84        | TYPE _ => None;   (*Typically (if thy doesn't include Numeral)
85 			     Undeclared type constructor "Numeral.bin"*)
87 end;